nLab normal bundle

Definition

Definition

For $i : X \hookrightarrow Y$ an immersion (notably an embedding) of smooth manifolds, the normal bundle of $X$ in $Y$ relative to $i$ is the vector bundle

$$N_i X \to X$$

defined as the fiberwise quotient bundle

$$N_i X =\frac{i^* T Y}{T X}$$

If $Y$ is equipped with Riemannian metric structure then this is equivalently the space of tangent vectors to $Y$ which are normal vectors to the tangent vectors to $X$, whence the name.

The pullback $i^* T Y$ can be of course interpreted as the restriction $T Y|_X$. The normal bundle is fiberwise the quotient of the fiber of the tangent bundle of $Y$ by the fiber of the tangent bundle of $X$: for $x \in X$

$$N_i (X)_x = T_{i(x)}Y/T_x(X) \,.$$

The dual notion is that of conormal bundle. The notion also makes sense for some other contexts, e.g. for smooth algebraic varieties.

Remark

There is always an isomorphism

$$T X \oplus N_i X \simeq T Y|_X \,,$$

but it is not canonically given, hence in particular cannot it general be chosen naturally. Except if certain additional structure is given:

If $Y$ is equipped with a Riemannian metric $g$, then we may identify the normal bundle with the bundle of vectors that are orthogonal to (“normal to”) the vectors in $T X$:

$$N_i(X) \simeq (T X)^\perp := \{ (x,v) \in T Y|_{X} | \forall w \in T_x X : g(v,w) = 0 \} \,.$$

Let $M^n$ be a smooth compact $n$-dimensional manifold without boundary, then the question of triviality of the normal bundle for an embedding $M^n\hookrightarrow \mathbf{R}^{n+r}$ for sufficiently large $r$ does not depend on the embedding. For this one uses the fact that any two such embeddings are regularly homotopic (this means the existence of a smooth homotopy $H(x,t)$ which is immersion for every $t \in [0,1]$ and which induces on the level of differentials a homotopy for the tangent bundles) and that any two regular homotopies are themselves homotopic through regular homotopies leaving end points fixed. Then one just uses the homotopy invariance of vector bundles. Thus, if $M^n$ admits an embedding into $\mathbf{R}^{n+r}$ with a trivial normal bundle then one says that $M^n$ has a stably trivial normal bundle. In that case, if $M^n_+$ is the union of $M$ with a disjoint base point, then there is a homeomorphism $T (M^n\times \mathbf{R}^{r})\cong \Sigma^r M^n_+$ where $\Sigma^r$ denotes the $r$-fold (reduced) suspension of based spaces $(S^r\times M^n_+)/(S^r\wedge M_+)$.

Related concepts

• Normal bundle plays a central role for instance in the theory of fiber integration by means of Pontrjagin-Thom collapse maps.

• tubular neighbourhood, conormal bundle

Literature

• Victor Snaith, Stable homotopy around the Arf-Kervaire invariant, Birkhauser 2009